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d) 10
c) 16
c) 32
d) 16
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Straight Objective type questions :
9.
The least 2 digit number that has 2, 5 and 8 as factors is
a) 80
b) 40
10. The sum of divisors of 16 is
a) 13
b) 31
11. The sum of divisors of 26 is
a) 24
b) 42
12. Statement (A): The product of factors of prime number is itself.
Statement (B) : (1184, 1210) and (5020, 5564) are the pairs of amicable numbers
a) Both A and B are true
b) Both A and B are false
c) A is true and B is false
d) A is false and B is true
c) 52
d) 32
Answers
ANSWER
Number of ways of expressing a given number as a product of two factors
N=ap.bq.cr where a,b,c are prime factors of N and p,q,r are positive integers) can be expressed as the product of two factors in different ways. The number of ways in which this can be done is given by the expression =1/2{(p+1)(q+1)(r+1)} If p,q,r etc. are all even, then the product (p+1)(q+1)(r+1)... becomes odd and the above rule will not be valid.
If p,q,r... are all even, it means that N is a perfect square.
So, to find out the number of ways in which a perfect square can be expressed as a product of 2 factors, we have the following 2 rules
(1) As a product of two DIFFERENT factors is (1/2){(p+1)(q+1)(r+1)...−1} ways (excluding X)
(2) As a product of two factors (including X) is (1/2){(p+1)(q+(r+1).+1} way
When n=2160=2
4
×3
3
×5
d=(4+1)(3+1)(1+1)=40
We have 40 divisors. So the Number of ways in which this can be written as a product of two numbers =
2
(4+1)(3+1)(1+1)
=
2
40
=20